Elliptic curve isogeny class with Cremona label 53371a (LMFDB label 53371.b)

• Label: 53371a
• Number of curves: $3$
• Conductor: $53371$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 53371a have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(19\)	\(1 - T\)
\(53\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 + 2 T + 2 T^{2}\)	1.2.c
\(3\)	\( 1 + 3 T^{2}\)	1.3.a
\(5\)	\( 1 - 3 T + 5 T^{2}\)	1.5.ad
\(7\)	\( 1 + T + 7 T^{2}\)	1.7.b
\(11\)	\( 1 - 3 T + 11 T^{2}\)	1.11.ad
\(13\)	\( 1 + 13 T^{2}\)	1.13.a
\(17\)	\( 1 + 3 T + 17 T^{2}\)	1.17.d
\(23\)	\( 1 - 8 T + 23 T^{2}\)	1.23.ai
\(29\)	\( 1 - 2 T + 29 T^{2}\)	1.29.ac
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 53371a do not have complex multiplication.

Modular form 53371.2.a.a

Isogeny matrix

The \((i,j)\)-th entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 53371a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
53371.b3	53371a1	\([0, -1, 1, 1873, -2003]\)	\(32768/19\)	\(-421122861451\)	\([]\)	\(48048\)	\(0.91997\)	\(\Gamma_0(N)\)-optimal
53371.b2	53371a2	\([0, -1, 1, -26217, -1729538]\)	\(-89915392/6859\)	\(-152025352983811\)	\([]\)	\(144144\)	\(1.4693\)
53371.b1	53371a3	\([0, -1, 1, -2161057, -1222057453]\)	\(-50357871050752/19\)	\(-421122861451\)	\([]\)	\(432432\)	\(2.0186\)